A note on projective Levi flats and minimal sets of algebraic foliations

Lins Neto, Alcides

Annales de l'Institut Fourier, Tome 49 (1999) no. 4, pp. 1369-1385.

Dans cet article on démontre qu’un feuilletage holomorphe de codimension un dans $ℂ{\mathbb{P}}^n,n\ge 3$, n’a pas de minimaux non triviaux. On démontre aussi que pour $n\ge 3$, il n’existe pas de surfaces de Levi plates, analytiques réelles, dans $ℂ{\mathbb{P}}^n$.

In this paper we prove that holomorphic codimension one singular foliations on $ℂ{\mathbb{P}}^n,n\ge 3$ have no non trivial minimal sets. We prove also that for $n\ge 3$, there is no real analytic Levi flat hypersurface in $ℂ{\mathbb{P}}^n$.

[AL-N] A. Lins Neto, Algebraic solutions of polynomial differential equations and foliations in dimension two, Springer Lecture Notes, 1345 (1988), 192-232. | MR | Zbl

[BB] P. Baum, R. Bott, On the zeroes of meromorphic vector fields, Essais en l'honneur de De Rham, (1970) 29-47. | MR | Zbl

[CLS1] C. Camacho, A. Lins Neto and P. Sad, Minimal sets of foliations on complex projective spaces, Publ. Math. IHES, 68 (1988) 187-203. | EuDML | Numdam | MR | Zbl

[CLS2] C. Camacho, A. Lins Neto and P. Sad, Foliations with algebraic limit sets, Ann. of Math., 135 (1992) 429-446. | MR | Zbl

[E] G. Elencwajg, Pseudo-convexité locale dans les variétés Kahlériennes, Ann. Inst. Fourier, 25-2 (1975), 295-314. | EuDML | Numdam | MR | Zbl

[G] M. Greenberg, Lectures on Algebraic Topology, W. A. Benjamin inc., 1967. | MR | Zbl

[H] M. Hirsh, Differential Topology, Springer Verlag, N.Y., 1976. | Zbl

[Ha] A. Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa, serie 3, vol. 16 (1962), 367-397. | EuDML | Numdam | MR | Zbl

[HL] G. M. Henkin and J. Leiterer, Theory of Functions on Complex Manifolds, Birkhäuser, 1984. | Zbl

[M] J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963. | Zbl

[MB] M. Brunella, Some remarks on indices of holomorphic vector fields, Prépublication 97, Université de Bourgogne (1996). | Zbl

[S] Y. T. Siu, Techniques of extension of analytic objects, Marcel Dekker Inc., New-York, 1974. | Zbl

[Sm] S. Smale, On gradient dynamical systems, Ann. of Math., 74 (1961). | Zbl

[ST] Y.T. Siu and G. Trautmann, Gap-sheaves and extension of coherent analytic subsheaves, Lect. Notes in Math., 172 (1971). | MR | Zbl

[T] A. Takeuchi, Domaines pseudo-convexes sur les variétés Kahlériennes, Jour. Math. Kyoto University, 6-3 (1967), 323-357. | MR | Zbl

[To] G. Tomassini, Tracce delle funzioni olomorfe sulle sottovarietà analitiche reali d'una varietà complessa, Ann. Scuola Norm. Sup. Pisa, (1966), 31-43. | Numdam | MR | Zbl